A recent book by Gregory Treverton, a risk analyst at the RAND Corporation whose work we have found highly valuable, contains a curious reflection:

When I spoke about the terrorist threat, especially in the first years after 2001, I was often asked what people could do to protect their family and home. I usually responded by giving the analyst's answer, what I labeled "the RAND answer." Anyone's probability of being killed by a terrorist today was essentially zero and would be tomorrow, barring a major discontinuity. So, they should do nothing. It is not surprising that the answer was hardly satisfying, and I did not regard it as such.

From this experience, he concluded, "People want information, but the challenge for government is to warn without terrifying."

It is not clear why anyone should find his observation unsatisfying since it simply puts the terrorist threat in general and in personal context, suggesting that excessive alarm about the issue is scarcely called for. It is, one might suspect, exactly the kind of accurate, reassuring, adult, and nonterrifying information people have been yearning for. And it deals frontally with a key issue in risk assessment: evaluating the likelihood of a terrorist attack.

Treverton's "RAND answer," calmly (and accurately) detailing the likelihood of the terrorist hazard and putting it in reasonable context, has scarcely ever been duplicated by politicians and officials in charge of providing public safety. Instead the awkward problem of dealing with exceedingly low probabilities has been finessed—and questionable expenditures accordingly justified—by stratagems that in various ways embrace a form of risk aversion that can be called "probability neglect."

Legal scholar and White House official Cass Sunstein, who seems to have invented the phrase, "probability neglect," assesses the version of the phenomena that comes into being when "emotions are intensely engaged," as they were after 9/11. Under that circumstance, he argues, "people's attention is focused on the bad outcome itself, and they are inattentive to the fact that it is unlikely to occur." Moreover, they are inclined to "demand a substantial governmental response—even if the magnitude of the risk does not warrant the response."

Playing to this demand, government officials are inclined to focus on worst-case scenarios, presumably in the knowledge, following Sunstein's insight, that this can emotionally justify just about any expenditure no matter how unlikely the prospect the dire event will actually take place.

Analyst Bruce Schneier has written penetratingly of worst-case thinking. He points out that it "involves imagining the worst possible outcome and then acting as if it were a certainty. It substitutes imagination for thinking, speculation for risk analysis, and fear for reason. It fosters powerlessness and vulnerability and magnifies social paralysis. And it makes us more vulnerable to the effects of terrorism."

Another technique is simply to rank relative risk while neglecting to determine the actual magnitude of the risk. It may be true that New York is more likely to be struck by a terrorist than, say, Columbus, Ohio. But it is also more likely to be struck by a tsunami, and not only in Hollywood disaster thrillers. Before spending a lot of money protecting New York from a tsunami, we need to get some sort of sense about what the likelihood of that event actually is, not simply how the risk compares to that borne by other cities. And the same goes for terrorism.

There is also a tendency to inflate the importance of potential terrorist targets. Thus, nearly half of American federal homeland security expenditure is devoted to protecting what the Department of Homeland Security and various presidential and congressional reports and directives rather extravagantly call "critical infrastructure" and "key resources"— assets whose loss would have a "debilitating effect on security, national economic security, public health or safety, or any combination thereof" or are "essential to the minimal operations of the economy or government." It is difficult to imagine what a terrorist group armed with anything less than a massive thermonuclear arsenal could do to hamper such "minimal operations." The terrorist attacks of 9/11 were by far the most damaging in history, yet, even though several major commercial buildings were demolished, both the economy and government continued to function at considerably above the "minimal" level.

A final, and very important, stratagem is to fail to assess, or massively to inflate, the capacities of the terrorists, and therefore by inference both the likelihood they will attack and the consequences of that attack. This is something that should be of absolutely key importance yet, in its big national infrastructure protection report of 2009, the DHS devotes only two paragraphs to describing the nature of the "terrorist adversary." Moreover, none of this fleeting discussion shows any depth, and the report prefers instead to spew out adjectives like "relentless," "patient," and "flexible," terms that scarcely characterize the vast majority of potential terrorists.

The ultimate in such thinking—common during the administration of George W. Bush and continued more sporadically in the administration of his successor, Barack Obama—is to characterize the terrorist threat as "existential." In 2008, Homeland Security Secretary Michael Chertoff even proclaimed the "struggle" against terrorism to be a "significant existential" one—carefully differentiating it, apparently, from all those insignificant existential struggles Americans have waged in the past. Rather amazingly, such extreme expressions, which if accepted as valid can close off all judicious evaluation of the problem, have only rarely been called into question.

In stark contrast, Glenn Carle, a 23‑year veteran of the Central Intelligence Agency, where he was deputy national intelligence officer for transnational threats, has warned, "We must not take fright at the specter our leaders have exaggerated. In fact, we must see jihadists for the small, lethal, disjointed and miserable opponents that they are." Al-Qaeda "has only a handful of individuals capable of planning, organizing and leading a terrorist organization," and although they have threatened attacks, "its capabilities are far inferior to its desires."

As for the domestic situation, the FBI and other investigative agencies have been unable to uncover a single true al‑Qaeda sleeper cell anywhere in the United States. Indeed, they have been scarcely able to unearth anyone who might even be deemed to have a "connection" to the diabolical group.

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Assessing the threat from homegrown Islamist terrorists, Brian Jenkins stresses that their number is "tiny," representing one out of every 30,000 Muslims in the United States. This "very low level" of recruitment finds very little support in the Muslim community at large: "they are not Mao's guerrillas swimming in a friendly sea." Given this situation, concludes Jenkins, what is to be anticipated is "tiny conspiracies, lone gunmen, one-off attacks rather than sustained terrorist campaigns." In the meantime, note other researchers, Muslim extremists have been responsible for 1/50th of 1 percent of the homicides committed in the United States since 9/11.

Because terrorism of a considerably destructive nature can be perpetrated by a very small number of people, or even by a single individual, the fact that terrorists are few in number does not mean there is no problem, and from time to time some of these people may actually manage to do some harm, though in most cases their capacities and schemes—or alleged schemes—seem to be far less dangerous than initial press reports suggest.

The situation seems scarcely different in Europe and other Western locations. Political scientist Michael Kenney has interviewed dozens of officials and intelligence agents and analyzed court documents. He finds that, in sharp contrast with the boilerplate characterizations favored by the DHS, Islamic militants there are operationally unsophisticated, short on know‑how, prone to make mistakes, poor at planning, and limited in their capacity to learn.

Indeed, since 9/11 no terrorist entity within the United States has been able to detonate even a simple bomb. And, except for the London Underground bombings of 2005, neither has any in the United Kingdom.

One can also focus on the kind of terrorism that really concerns people in the developed world by tallying the violence committed by Muslim extremists outside of war zones, whether that violence is perpetrated by domestic Islamist terrorists or by ones with international connections. Three publications from think tanks have independently provided lists or tallies of such violence committed in the several years after the 9/11 attacks. The lists include not only attacks by al‑Qaeda but also those by its imitators, enthusiasts, look‑alikes, and wannabes, as well as ones by groups with no apparent connection to it whatever.

Although these tallies make for grim reading, the total number of people killed in the years after 9/11 by Muslim extremists outside of war zones comes to some 200 to 300 per year. That, of course, is 200 to 300 too many, but it hardly suggests that the destructive capacities of the terrorists are monumental. For comparison, during the same period more people—320 per year—drowned in bathtubs in the United States alone. Or there is another, rather unpleasant comparison. Increased delays and added costs at U.S. airports due to new security procedures provide incentive for many short‑haul passengers to drive to their destination rather than flying, and, since driving is far riskier than air travel, the extra automobile traffic generated has been estimated in one study to result in 500 or more extra road fatalities per year.

The fact that the public has difficulties with probabilities when emotions are involved does not relieve those in charge of the requirement, even the duty, to make decisions about the expenditures of vast quantities of public monies in a responsible manner. In the end, one might darkly suspect, various versions of probability neglect are grasped because, if realistic probabilities that a given target would be struck by terrorists were multiplied into the risk calculation and if the costs of protection from unlikely threats were sensibly calculated following standard procedures, it would be found that vast amounts of money have been misspent.

Tomorrow: We'd have to foil 1,667 Times Square-style attacks every year to justify current spending on homeland security.

John Mueller is a political scientist at Ohio State University and a senior fellow at the Cato Institute.